When teachers reward behavior, we try to motivate students to do something, for instance, to improve an academic skill or to behave in a different way. The expectation is that the reward will strengthen the skill or behavior we want; if the child produces the “right” response or behavior (defined by the teacher), we reward, if not, we withhold. According to the American Heritage College Dictionary (Fourth Edition), rewards are something given in recompense for worthy behavior, most specifically (psychology definition), the return for performance of a desired behavior. The same source defines “bribe” as something offered or given to influence a person’s view or conduct. The extended definition identifies “bribes” as something that serve to influence or persuade. Based strictly on these two definitions, only one thing is clear to me: the boundaries between rewarding children and bribing children are blurred, and they even may be overlapping. Simply put, both rewards and bribes can be interpreted as an attempt to influence and change children’s behavior. In addition, bribery may involve the use of rewards. The best defense in using reinforcement procedures in the classroom, including tangible and non-tangible rewards, is that, when teachers reward behavior, we are trying to make our students more competent or help them change some aspect of their behavior in a positive way. Some teachers like using rewards; others object to their use, sometimes strongly. I would love to read pros and cons on this issue.
Related Article:
Should Teachers Give Rewards to Students for Good Behavior? A Psycho-Educational Perspective
http://thepsychoeducationalteacher.blogspot.com/2010/11/should-teachers-...

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The problem with rewards is that they take away the sense of achievement, and substitute it with the joy of the extrinsic reward (which often doesn't even relate to the achievement). This is not how we build accountability. It only builds obedience (and quite low level in that, too, please remember what Kohlberg wrote about stages of morality).

Furthermore, in learning situations rewards take away the logical path: cause-effect-consequence, because the learning process gets interrupted by this extrinsic reward (or punishment) and the student never experiences the complete process. This harms their learning and caters for thinking intelligence being fixed instead of something we learn and which grows through our everyday experiences.

I won't even start to discuss bribery here. I just want to note how both bribing and rewarding are ancient tools for using power over others, and they really don't belong to schools, where the basic ideology should circle around the best practices for empowering students to learn.

To differentiate between bribes and rewards, it all depends on the timing.
Scenario 1: I will give you this object if you will stay quiet for the entire class.
In this case, the extrinsic object is being given before the behavior is exhibited. That makes it a bribe.
Scenario 2: You are being extremely respectful and helpful in my classroom, here is a token of my appreciation.
In this case, the reward is being given after the behavior has been demonstrated. That makes it a reward.

In both cases, extrinsic objects are being used for behavioral support and not as a form of reward for doing classwork.

I've always looked as rewards as positive reinforcement of almost any kind, and bribes as being coercive in nature. In other words, when a child does something well, or proximately well, the reward (anything from praise to stickers to "incentives") reinforce the motivation to repeat the behavior. changing incentives or what people are working towards keeps the motivation going as well. Bribes are payment for doing something for your benefit, not the students.
We used tokens/points/incentive charts to help my son learn how to read; it worked like a charm, especially since once the skill is learned, it's not forgotten.
We do jobs because we get paid; is that bribery or coercion, or positive reinforcement?
I think the "rule" if there is one, is to remember all of theSkinner behavioralist rules on positive and negative reinforcement, and figuring out whether you have an incentive or bribe becomes pretty clear.
ps. We all want kids to be intrinsically motivated to learn, yet one of my kids, who is prone to watch YouTube videos and learn more than he needs to know about subjects at hand frustrates the crud out of his teachers, because the learning and exploration is more important to him than memorization for performance on a test. We say we want intrinsic motivation to learn, but we don't know what to do with the kids who don't bow to extrinsic pressure like grades.

I agree with a lot of the comments offered above. One of my big take-aways from teaching was the importance of transitioning from extrinsic to intrinsic rewards. Meet your kids where they are with the rewards that motivate them, but do so with the goal of helping them get to the point where these behaviors that you want to encourage (maybe it's homework completion or attendance) are done for the sake of doing them. Aren't rewards a means to an end?

One of my other big takeaways was making the rewards deceptively academic. I taught in Washington, DC. If my students did really well on a unit, I bought them donuts and took them on a weekend field trip to a museum. Not every student responded to this, but, for those that did, it was a powerful reminder that incentives can be academic.